Hermite's approach to Abelian integrals revisited

TL;DR

This paper generalizes Hermite's classical results on Abelian integrals by establishing a new linear independence criterion for Lauricella hypergeometric series over algebraic number fields, using Padé approximations and determinant non-vanishing techniques.

Contribution

It introduces a novel linear independence criterion for Lauricella hypergeometric series, extending Hermite's work on Abelian integrals with new methods involving Padé approximations.

Findings

Established linear independence over complex and p-adic fields.

Extended Hermite's theorem to Lauricella hypergeometric series.

Proved non-vanishing of determinants for Padé approximants.

Abstract

In this article, we establish a new linear independence criterion for the values of certain {\it Lauricella hypergeometric series} $F_D$ with rational parameters, in both the complex and $p$-adic settings, over an algebraic number field. This result generalizes a theorem of C.~Hermite \cite{Hermite} on the linear independence of certain Abelian integrals. Our proof relies on explicit Pad\'{e} type approximations to solutions of a reducible Jordan-Pochhammer differential equation, which extends the Pad\'{e} approximations for certain Abelian integrals in \cite{Hermite}. The main novelty of our approach lies in the proof of the non-vanishing of the determinants associated with these Pad\'{e} type approximants.